Second-order PDEs in four dimensions with half-flat conformal structure
journal contributionposted on 06.01.2020, 15:36 by Sobhi BerjawiSobhi Berjawi, Evgeny FerapontovEvgeny Ferapontov, Boris Kruglikov, Vladimir NovikovVladimir Novikov
We study second-order partial differential equations (PDEs) in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge–Ampère property. Since half-flatness of the conformal structure is equivalent to the existence of a non-trivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable dispersionless PDEs in dimensions four and higher are of Monge–Ampère type. Some partial classification results of Monge–Ampère equations in four dimensions with half-flat conformal structure are also obtained.
EPSRC grant EP/N031369/1
Trond Mohn Foundation
Tromsø Research Foundation
- Mathematical Sciences